To factor the polynomial [tex]\( x^4 - 3x^2 + 1 \)[/tex], we look for a way to express it as a product of simpler polynomials. Let's go through the steps in detail:
1. Identify the Polynomial:
We start with the polynomial [tex]\( P(x) = x^4 - 3x^2 + 1 \)[/tex].
2. Attempt to Find Factors:
We recognize that this polynomial is a quadratic in terms of [tex]\( x^2 \)[/tex]. We can think of [tex]\( x^2 \)[/tex] as a single variable and rewrite the polynomial as:
[tex]\[
P(x) = (x^2)^2 - 3(x^2) + 1
\][/tex]
Let [tex]\( u = x^2 \)[/tex]. Then the polynomial becomes:
[tex]\[
P(u) = u^2 - 3u + 1
\][/tex]
3. Factorize the Quadratic in Terms of [tex]\( u \)[/tex]:
Now we need to factor the quadratic expression [tex]\( u^2 - 3u + 1 \)[/tex]. We look for two numbers whose product is 1 (constant term) and whose sum is -3 (coefficient of the middle term). Unlike typical elementary factorization, this expression factors nicely into special forms:
[tex]\[
u^2 - 3u + 1 = (u - a)(u - b)
\][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] satisfy the quadratic equation [tex]\( t^2 - 3t + 1 = 0 \)[/tex].
4. Find Roots of the Quadratic:
Solve the quadratic equation:
[tex]\[
t^2 - 3t + 1 = 0
\][/tex]
The roots are given by the quadratic formula:
[tex]\[
t = \frac{3 \pm \sqrt{9 - 4}}{2}
= \frac{3 \pm \sqrt{5}}{2}
\][/tex]
Let these roots be [tex]\( r_1 = \frac{3 + \sqrt{5}}{2} \)[/tex] and [tex]\( r_2 = \frac{3 - \sqrt{5}}{2} \)[/tex].
5. Return to the Original Variable:
Recall [tex]\( u = x^2 \)[/tex], hence the polynomial [tex]\( x^4 - 3x^2 + 1 \)[/tex] can be written considering roots:
[tex]\[
(x^2 - r_1)(x^2 - r_2)
\][/tex]
However, for clarity, we'll use integral coefficients.
6. Final Factorization:
Hence, combining the terms we have:
[tex]\[
\boxed{(x^2 - x - 1)(x^2 + x - 1)}
\][/tex]
This is the final factoring of the given polynomial [tex]\( x^4 - 3x^2 + 1 \)[/tex].
